Relations among the squares of the generators of the braid group
type: Item
instance of: scholarly article
title: Relations among the squares of the generators of the braid group
zbMATH Open document ID: 0822.20040
DOI: 10.1007/BF01232254
author: Donald J. Collins
published in: Inventiones Mathematicae
publication date: 1995-10-05T00:00:00Z
full work available at URL: https://eudml.org/doc/144225
review text: Let \(M=(m_{ij})\) denote an \(n \times n\) Coxeter matrix \((m_{ij}=m_{ji} \in \mathbb{Z}^{\geq 2} \cup \{\infty\}\) for \(i \neq j\) and \(m_{ii}=1\)). Let \(A(M)\) be an Artin group with generating set \(\{a_ 1, \dots, a_ n\}\) and for each \(i \neq j\) a relation \(a_ i a_ j a_ i \dots=a_ j a_ i a_ j \dots\), where both sides are words of length \(m_{ij}\). If \(m_{ij}=2\) then \(a^ 2_ i\) and \(a^ 2_ j\) commute. The Tits conjecture [stated in: \textit{K. I. Appel} and \textit{P. E. Schupp}, Invent. Math. 72, 201-220 (1983; Zbl 0536.20019)] says that for an arbitrary Artin group the only relations between the \(a^ 2_ i\) are these obvious commutator relations. The author proves the Tits conjecture for the braid groups.
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 703859
zbMATH Keywords: subgroups of braid groups
zbMATH Keywords: Artin groups
zbMATH Keywords: generating sets
zbMATH Keywords: relations
zbMATH Keywords: Tits conjecture
zbMATH Keywords: commutator relations
reviewed by: Andrei M. Akimenkov
MaRDI profile type: Publication
cites work: Q5748589
cites work: On Tit's conjecture and other questions concerning Artin and generalized Artin groups
cites work: Normalisateurs de tores. I: Groupes de Coxeter etendus
description: scientific article; zbMATH DE number 703859